Uniform existence of the integrated density of states for combinatorial and metric graphs over Zd

We give an overview and extension of recent results on ergodic random Schrodinger operators for models on Z(d). The operators we consider are defined oil combinatorial or metric graphs, with random potentials, random boundary conditions and random metrics taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable, at least locally. This limit, the integrated density of states (IDS), can be expressed by a closed Shubin-Pastur type trace formula. The set of points of increase of the IDS supports title spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. This applies to several examples, including various periodic operators and percolation models.